Feb 24, 2015 - called enzyme based logic) which code for two-inputs logic gates and mimic the stochastic AND ( ..... molecule and the free energy of the binding process. ..... energy transfer (FRET) to decreasing concentrations (in mM) of a-.
OPEN SUBJECT AREAS: BIOPHYSICAL CHEMISTRY
Notes on stochastic (bio)-logic gates: computing with allosteric cooperativity Elena Agliari1, Matteo Altavilla2, Adriano Barra1, Lorenzo Dello Schiavo2 & Evgeny Katz3
BIOCHEMICAL NETWORKS COMPUTATIONAL BIOPHYSICS
1 3
Received 7 November 2014 Accepted 24 February 2015 Published 15 May 2015
Correspondence and requests for materials should be addressed to A.B. (adriano.barra@ roma1.infn.it)
Dipartimento di Fisica, Sapienza Universita di Roma, Italy, 2Dipartimento di Matematica, Sapienza Universita di Roma, Italy, Department of Chemistry and Biomolecular Science, Clarkson University, New York, USA.
Recent experimental breakthroughs have finally allowed to implement in-vitro reaction kinetics (the so called enzyme based logic) which code for two-inputs logic gates and mimic the stochastic AND (and NAND) as well as the stochastic OR (and NOR). This accomplishment, together with the already-known single-input gates (performing as YES and NOT), provides a logic base and paves the way to the development of powerful biotechnological devices. However, as biochemical systems are always affected by the presence of noise (e.g. thermal), standard logic is not the correct theoretical reference framework, rather we show that statistical mechanics can work for this scope: here we formulate a complete statistical mechanical description of the Monod-Wyman-Changeaux allosteric model for both single and double ligand systems, with the purpose of exploring their practical capabilities to express noisy logical operators and/or perform stochastic logical operations. Mixing statistical mechanics with logics, and testing quantitatively the resulting findings on the available biochemical data, we successfully revise the concept of cooperativity (and anti-cooperativity) for allosteric systems, with particular emphasis on its computational capabilities, the related ranges and scaling of the involved parameters and its differences with classical cooperativity (and anti-cooperativity).
C
ell’s life is based on a hierarchical and modular organization of interactions among its molecules1: a functional module is defined as a discrete ensemble of reactions whose functions are separable from those of other molecules. Such a separation can be of spatial origin (processes ruled by short range interactions) or of chemical origin (processes requiring specific interactions)2. The latter, i.e., chemical specificity, is at the basis of biological information processing3,4. A paradigmatic example of this is the signal transduction pathway of the so called two signal model in immunology by which an effector lymphocyte needs two signals (both integrated on its membrane’s highly-specific receptors in a close temporal interval) to get active5: these signals are the presence of the antigen (via the complex MHC-TCR) and the consensus of an helper-cell (via CD40 and an eliciting cytokine); this constitutes a biological, stochastic AND gate6. We added the adjective stochastic because, quoting Germain, ‘‘as one dissects the immune system at finer and finer levels of resolution, there is actually a decreasing predictability in the behavior of any particular unit of function’’, furthermore, ‘‘no individual cell requires two signals (¼) rather, the probability that many cells will divide more often is increased by co-stimulation’’7: as a result, standard logic (where operations follow a deterministic chain) plays as the ideal reference framework, while an operative one -a stochastic formulation of logic- should take into account the presence of noise too. Beyond countless natural examples, biologic gates have been realized even experimentally, see e.g. Refs. 8–18, the ultimate goal being the realization of stochastic, yet controllable, biological circuits19–22. Such striking outcomes also arouse a great theoretical attention aimed to develop a self-contained framework able to highlight their potentialities and suggest possible developments. In particular, statistical mechanics has proved to be a proper candidate tool for unveiling biological complexity: in the past two decades statistical mechanics has been applied to investigate intra-cellular (e.g. metabolomics23,24, proteinomics25,26) as well as extra-cellular (e.g. neural networks3,27, immune networks28,29) systems. Also, statistical mechanics intrinsically offers a partially-random scaffold which is the ideal setting for a stochastic logic gate theory. Another route to unveil the spontaneous information processing capabilities of biological matters is naturally constituted by information theory (see e.g. Refs. 30, 31 and references therein): remarkably, statistical mechanics and information theory (see the seminal works by Khinchin32,33, and by Jaynes34,35) and, in turn, information theory and logics (see the seminal works by Von Neumann36, and by Chaitin37) have been highlighted to be deeply connected. Therefore, it is not surprising that even in the quantitative modeling of biological phenomena these two routes are not conflicting but, rather, complementary. SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
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www.nature.com/scientificreports In this work, we will use the former (statistical mechanics) to describe a huge variety of biochemical allosteric reactions, and then, through the latter (mathematical logic), we will show how these reactions naturally encode stochastic versions of boolean gates and are thus capable of noisy information processing. We will especially focus on allosteric reactions (as those of Koshland, Nemethy and Filmer (KNF)38 and Monod-WymanChangeaux (MWC)39) as they play a major role in enzymatic processes for which a great amount of experimental data is nowadays available. However, classical reaction kinetics (i.e. those coded by Hill, Adair, etc.40) can also perform logical calculations and they have been set in a statistical mechanical scaffold in Ref. 19: along the paper we will deepen the crucial differences between the two types of kinetics -allosteric cooperativity versus standard cooperativity- when framed within statistical mechanics.
Results In the case of allosteric receptors, several models have been introduced. Many of these assume that a receptor can exist in either an active or inactive state, and that binding of a ligand biases the receptor to one of the two states. In particular, in the Monod-WymanChangeaux (MWC) model, ligand-bound receptors can be in either state, but coupled receptors switch between states in synchrony. Beyond that pioneering work, several models able to provide qualitative and quantitative descriptions of binding phenomena have been further introduced in the Literature, as e.g. the sequential model by Koshland, Nemethy and Filmer (KNF). Here we consider MWC-like kinetics, and we reformulate it into a statistical mechanical framework. We start by introducing terminology and parameters for mono-receptor/mono-ligand systems (playing for single input gates as YES and NOT) and then we expand such a scenario in order to account for the kinetics of more complex systems (double-receptors/double-ligands, as those will play for two-input gates as AND, NAND, OR, NOR). The plan is as follows: Once introduced the microscopic settings (e.g., the occupancy states si, i g (1, ¼, n) of n receptors and the dissociation energy h), we define the Hamiltonian functions H(s, h) coding for the chemical bindings; then -being b the thermal noise b 5 1/kBT (where kB is the Boltzmann constant and T represents the temperature) - we build the related Maxwell-Boltzmann probabilistic weights / exp[2bH(s, X h)]. The latter allows computing the parexpð{bH Þ, both for the active state ZA tition functions Z~ s and for the inactive ZI state; the ratios, pA 5 ZA/(ZA 1 ZI) and pI 5 ZI/(ZA 1 ZI) then return the probabilities of the active/inactive states as functions of the parameters (e.g. b, h, n). These probabilities are first analyzed from a logic perspective in order to show how they can account for boolean gates and then used to successfully fit the outcomes of the experiments on enzyme based logic. This route, although rather lengthy, shows why allosteric mechanisms share similar behaviors with those of classical cooperativity, but, at the same time, clearly reveals deep differences between these phenomena. System description. Specifically, we start considering a system built of several molecules, each displaying one or more receptors. Each receptor exhibits multiple binding sites where a ligand can reversibly bind, and which can exist in two different states (i.e. active and inactive). In general the receptors exhibited by a given molecule can differ in e.g., the number of binding sites, the affinity with ligands, etc. As we are building a theory for single and double input gates, in the following, we will focus on simple systems where receptors can house only one or two kinds of binding sites, as exemplified in Fig. 1. The simplest system we consider is made of a set of receptors of the same kind and in the presence of a unique ligand (see panel a in Fig. 1). More precisely, each receptor is constituted by n functionally SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
Figure 1 | This scheme summarizes the kind of systems we are considering here: Mono-receptor/Mono-ligand (a), Mono-receptor/Double-ligand (b) and Double-receptor/Double-ligand (c). In this cartoon all molecules are shown as dimeric, but cases a and b also work with monomeric structures. In the Mono-receptor/Mono-ligand case only one kind of receptor and one kind of ligand (compatible with the receptor) are considered; in the Monoreceptor/Double-ligand case we still have one kind of receptor, but two different ligands both compatible with the receptor; in the Doublereceptor/Double-ligand case we consider molecules displaying two different receptors in the presence of two different ligands, each compatible with only one receptor. The kinetics of these systems is addressed in System description Section, while in Section Logical Operation they are shown to work as YES, OR, and AND logic gates. See also Ref. 42.
identical binding sites indexed by i, whose occupancy is given by a boolean vector s 5 {si}, i 5 1, ¼, n where si 5 1 (respectively 0) indicates that the binding site i is occupied (respectively vacant). As required by the all-or-none MWC model, each receptor is either active (T) or inactive (R); the receptor state is indicated by a boolean activation parameter a, (a 5 0, 1)41,42. In the absence of the ligand, the active and inactive states (which are assumed to be in equilibrium) differ in their chemical potential, whose delta, indicated by E, can, in principle, be either positive (favoring the inactive state) or negative (favoring the active state): note that, the presence of a difference E in energy between the active and inactive states implies an exponentially unbalanced ratio between their relative concentrations (ruled by the MaxwellBoltzmann weight). Given a system of receptor molecules in the absence of ligand and in equilibrium at a given temperature T, we pose the following assumptions: (a)
(b)
As both the active and inactive state may coexist, the composition of the system also depends on the parameter L ; L(b) . 0, namely the equilibrium constant at inverse temperature b (in proper units, namely setting the Boltzmann constant KB ; 1). Letting [R] be the total concentration of the receptors, [RA] (respectively [RI]) the concentration of the active (respectively inactive) receptors in the in absence of ligand, it is [R] 5 [RA] 1 [RI] and [RA] 5 L[RI]; For the sake of simplicity, binding sites of a mono-receptor are considered as functionally identical (as in the original model39).
In the absence of ligand, we also need to establish which of the two states (namely the active and inactive one) has a higher chemical potential. As shown in the Literature (see Ref. 41 and below) the choice is in general arbitrary (i.e. case dependent), hence we take both possibilities into account. We therefore consider two sets of mutually exclusive assumptions (the latter of which is denoted by a ‘‘prime’’ symbol). (c)
The active state has a higher chemical potential. Notice that, while this assumption is in contrast with the original MWC model39, the model itself is still self consistent as thoroughly explained in Ref. 41. The same conclusion may be drawn by the fact that, in the MWC paper, the opposite assumption is merely exploited for calculations. (i.e. E . 0), as e.g. in Refs. 41, 43, hence the inactive state must then be predominant (to minimize energy) (i.e. L=1); 2
SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
pA
pI
pA
T
a21
c21
L v0
KA KI KI ½S R
T
neat percentage activation enhancement probability of the inactive (relaxed) state, i.e. average concentration of the receptor in the inactive state probability of the active (tense) state, i.e. average concentration of the receptor in the active state
equilibrium constant of active/ inactive–state receptor system in absence of the ligand dissociation constants ratio
pA pA
c
a pI
(e9) The ligand is a suppressor, i.e. the presence of the ligand hindrances activation of the receptor. Therefore, the occupation of each receptor singularly increases the energy required for activation by a parameter Ew0.
e2h
AUT
KI KA ½S KI R
The ligand is an activator, i.e. the presence of the ligand enhances activation of the receptor. Therefore, the occupation of each receptor singularly decreases the energy required for activation by a parameter Ew0.
eE
(e)
L
Finally, a ligand can play as an activator (i.e. its presence enhances the receptor activation) as well as a suppressor (i.e. its presence hindrances receptor’s activation) depending on the chemical potential time by time associated to the chemical reaction under examination (assumptions (c)’s). This choice is dual and will be deepened later: to avoid trivial (i.e. static) behavior of the system, we have to set either
v0
receptor-ligand solution is homogeneous and isotropic. This mean-field-like assumption is actually a key assumption of all the approaches in modeling classical reaction kinetics, see e.g. Ref. 19.
e
(d)
Ref. 39
in accordance with the original presentation of MWC model (in Ref. 39, p. 90, microscopic dissociation constants of a ligand [¼] bound to a stereospecific site are considered, whose arithmetic weighted means we denote as global dissociation constants.). The dynamics of the receptor/ligand system is therefore determined by the variable [S] and the parameters KA, KI. Now, considering both the ligand and the receptor/ligand solution we assume that
MWC meaning
ð2Þ
Ref. 42
½S½R , h½RI Si
2E
KI :
Stat-Mech A-set
½S½R , h½RA Si
Table 1 | Correspondences with the parameters originally used in Ref. 39
KA :
eh
We can define the average concentration h[RI S]i of the inactive receptor/ligand complex in an analogous way, and we can then set
e{E
Stat-Mech, A’ -set
Ref. 42
Ref. 39
For a thorough comparison of these two alternative assumptions (and those of the original MWC) we refer to Tab. 1. For the sake of clarity we will from now on refer to the (c)-type assumptions as ‘‘assumptions A’’ and to the (c9)-type assumptions as ‘‘assumptions A’’’. We also refer to the A’ -set of assumptions as dual to assumptions A, where this terminology is introduced to match the one of mathematical logic and will be therefore explained in Section ‘‘Logical operations’’. All the assumptions without a dual one are taken to be part of both the assumption sets. Let us now discuss the case of a system of receptor molecules in the presence of ligand. Clearly, the behavior of the system is expected to depend on ligand’s concentration [S] and on the receptor state (i.e. either active or inactive). The dependence on the receptor state is formalized by introducing dissociation constants KA and KI for the receptor in the active and inactive state, respectively (see Ref. 42). Letting [(RAS)i] be the concentration of the receptor/ligand complex’s molecules which have exactly i occupied binding sites, we define the average concentration of the active receptor/ligand complex as n 1X i: ðRA SÞi : ð1Þ h½RA Si: n i~1
e
MWC meaning
(c9) The active state has a lower chemical potential (i.e. E9 ; 2E , 0) as e.g. in the original MWC model39, hence (still for minimum energy requirement) the active state must then be predominant (i.e. L?1).
2E9
AUT
equilibrium constant of active/ inactive-state receptor system in absence of the ligand (inverse) dissociation constants ratio inverse neat percentage activation enhancement probability of the inactive (relaxed) state, i.e. average concentration of the receptor in the inactive state probability of the active (tense) state, i.e. average concentration of the receptor in the active state
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3
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ZA ~
X
e{bHA ðsÞ
ð5Þ
e{bHI ðsÞ ,
ð6Þ
fsg
ZI ~
X fsg
while the total partition function Z is given by X e{bH ðs,aÞ ~ZA zZI : Z~
ð7Þ
fsg,fag
Figure 2 | This scheme summarizes the states and the weights of the simplest MWC molecule (that, in turn, codes for the YES gate). Having only one binding site (n 5 1), the number of possible states is four, from top to bottom: inactive and vacant, active and vacant, inactive and occupied, active and occupied. Each state corresponds to an energy, to a statistical mechanics weight and to a chemical weight. The energy is obtained by considering both the conformational degree of freedom of the molecule and the free energy of the binding process. The statistical mechanics weight is obtained according to the Boltzmann factor and the chemical weight is derived according to Tab. 1. See also Ref. 48.
Mono receptor/Mono ligand (MM) properties at equilibrium. Under assumptions A, any mono-receptor/mono-ligand system, built by n receptors [i g (1, ¼, n)], and whose occupancy is ruled by si 5 (0, 1), can be described by the following allosteric Hamiltonian function ! n n X X H ðs,aÞ~ E{E si azh si , ð3Þ i~1
ðs,1Þ
i~1
where we recall that E is the energy delta given by chemical potential, and h is the dissociation energy, namely the energy captured by a single binding site of the inactive state receptor by binding to a ligand molecule; the term in the brackets accounts for the fact that ligand acts as an activator since, for the active state (a 5 1) binding is energetically favored, while in the inactive state (a 5 0) the related term disappears in the Hamiltonian that reduces to the last term accounting for the bare association energy only. By the same reasoning under assumptions A’, we obtain ! n n X X si azh si , ð4Þ H ðs,aÞ~ {EzE i~1
A few remarks are in order here: – The summations in the partition function (7) account for the activation degree of freedom too. This means that the latter participate in thermalization or, in other words, that the intrinsic timescale for the dynamics of a is bounded from above by those of the s: this is consistent with the original MWC assumptions of synchronized switches among coupled receptors (the so called all-or-none behavior). – This model can be solved even at finite n, namely without the oversimplifying thermodynamic limit n R ‘. – All the energies can be expressed in units of the thermal energy kBT ; b21, hence, in order to avoid possible misunderstanding as T already addresses the tense molecular state and to keep notation as simple as possible, in the following we set b 5 1, thus forcing all aforementioned parameters and variables to be dimensionless. – As a consequence of the two previous remarks, the stochasticity is retained by the parameter n, such that for n R ‘ stochastic computing will collapse on the deterministic one (that is, classical logic) and, on the other side, the smaller n and the larger the noise affecting the system. Xn s, Now, focusing on ZA (as ZI is analogous), we define k~ i~0 i and we can therefore write X e{H ðs,1Þ ~ ð8Þ ZA ~
i~1
The main features of the mono-receptor/mono-ligand systems described above are summarized in Fig. 2. It is worth highlighting that the Hamiltonians (3) and X (4) do not s s : this include any two-body couplings, i.e. any term ! ij i j framework is intrinsically one-body in the statistical mechanical vocabulary and this has implications in biochemistry too. For instance, one-body models do not undergo phase transitions and, as the latter mirror ultrasensitive reactions in chemical kinetics19, those are ruled out by these Hamiltonians. Since the activation parameter is boolean, the receptor/ligand complex state may be considered regardless of the state of the receptor, by introducing the two Hamiltonians HA(s) ; H(s, 1) and HI(s) ; H(s, 0), defining the active and the inactive state energy, respectively. The corresponding partition functions are SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
~
n X
Ak e{ðE{kEÞ{hk ~e{E
k~0
n X
Ak ekðE{hÞ ,
ð9Þ
k~0
Xn where Ak denotes the number of times that the sum s turns i~1 i out to be equal to k. Noting that s is a binary vector, we get straight n forwardly that Ak ~ , and therefore k ! n X n kðE{hÞ {E ZA ~e e k k~0 ð10Þ ! n X n kðE{hÞ n{k {E :1 ~e e k k~0 h in ~e{E 1zeðE{hÞ :
ð11Þ
Analogously, ZI 5 (1 1 e2h)n. Thus, the probabilities pA and pI for the complex to be in the active or in the inactive state, respectively, are n e{E 1zeE{h ZA ðpA ÞMM ~ ~ {E , ð12Þ ZA zZI e ð1zeE{h Þn zð1ze{h Þn n 1ze{h ZI ðpI ÞMM ~ ~ : ZA zZI e{E ð1zeE{h Þn zð1ze{h Þn
ð13Þ
where the subscript MM stands for ‘‘Mono-Mono’’. 4
www.nature.com/scientificreports Correspondingly,
{E{h n
p’A MM ~
e 1ze ZA ~ E , ZA zZI e ð1ze{E{h Þn zð1ze{h Þn
ð14Þ
p’I MM ~
n 1ze{h ZI ~ E : ZA zZI e ð1ze{E{h Þn zð1ze{h Þn
ð15Þ
E
The interesting quantity to look at is (pA)MM, as it corresponds to of receptors in the active state and this is the concentration T expected to continuously increase (respectively decrease) with the percentage of activation enhancement (i.e. e2h, see Tab. 1) under assumptions A (respectively A’). We notice, though, that the original anyhow, pA (i.e. with pI) rather than T; model39 is concerned with R and pI carry the same information as they are complementary probabilities. Notably, the correspondence stated in Tab. 1 confirms the consequences of assumptions (c) and (e), that is, choosing E . 0 yields L , 1, while choosing Ew0 yields c . 1. In particular, according to the notation of Ref. 39, we have R~
ð1zaÞn , Lð1zcaÞn zð1zaÞn
ð16Þ
Lð1zcaÞn T~1{ R~ : Lð1zcaÞn zð1zaÞn
ð17Þ
Conclusions on the dual assumptions A’ are much the same and will not be repeated. Mono-receptor/Double-ligand (MD) properties at equilibrium. Under the assumptions of the previous section, any mono-receptor/doubleligand system, built by n receptors [i g (1, ¼, n)] and whose occupancy is ruled by si 5 (0, 1), can be described by the following allosteric Hamiltonian function ! n X X X H ðs,a,I,J Þ~ E{E si azh1 si zh2 sj , ð18Þ i~1
i[I
j[J
where, in contrast with the previous case described by eq. (3), two distinct ligands, whose dissociation energies are denoted by h1 and h2 respectively, are considered. More precisely, I and J are two subsets of {1, ¼, n} such that I\J~1, and they denote the sites linked to the first ligand and to the second ligand, respectively. To express this in formulae, we impose that I|J~findices such that si ~1g. As we did for the Mono-Mono case, the partition function coupled to the Hamiltonian (18) is given by X Z~ e{H ðs,aÞ ðs,aÞ
~
X
e{H ðs,0Þ z
ðs,0Þ
X
e{H ðs,1Þ
ð19Þ
ðs,1Þ
~ZI zZA :
ð20Þ
We focus on ZA, as ZI is analogous. Let us pose k1 5 jIj and k2 5 jJj, Xn notice that k~k1 zk2 ~ s , and write the sums explicitly as i~0 i ZA ~
X ðs,1Þ
e{H ðs,1Þ ~
n X k X
Ak,k1 e{ðE{kEÞ{h1 k1 {h2 ðk{k1 Þ ,
k~0 k1 ~0
ð21Þ
Xn s is where Ak,k1 denotes the number of times that the sum i~0 i equal to k, with the condition that k1 of the si’s belong to the set I. This quantity is rather tricky to calculate but can actually be rewritten in terms of multinomial coefficient (which counts the number of SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
ways we can choose k elements among n, with the condition that they are divided in groups of kj elements each). Then, we get n n Ak,k1 ~ ~ , ð22Þ k1 ,k{k1 k1 ,k2 in such a way that ZA can be rewritten (using k1 and k2) as Xn Xk1 zk2 n eðk1 zk2 ÞE{h1 k1 {h2 k2 ZA ~e{E k1 ~0 k1 zk2 ~0 k1 ,k2 Xn Xk1 zk2 n ek1 ðE{h1 Þ :ek2 ðE{h2 Þ ~e{E k1 ~0 k1 zk2 ~0 k1 ,k2 h in ~e{E 1zeðE{h1 Þ zeðE{h2 Þ ,
ð23Þ ð24Þ ð25Þ
where in the second passage we must consider a 1n{ðk1 zk2 Þ factor, which allows us to conclude the calculation, by simply expanding the trinomial. n Analogously, we obtain ZI ~ 1ze{h1 ze{h2 . Indeed, we have n e{E 1zeE{h1 zeE{h2 ðpA ÞMD ~ {E ð26Þ n n, e ð1zeE{h1 zeE{h2 Þ zð1ze{h1 ze{h2 Þ ðpI ÞMD ~
n 1ze{h1 ze{h2 n n: e{E ð1zeE{h1 zeE{h2 Þ zð1ze{h1 ze{h2 Þ
In a similar fashion, under assumptions A’, we obtain n eE 1ze{E{h1 ze{E{h2 p’A MD ~ E n n, e ð1ze{E{h1 ze{E{h2 Þ zð1ze{h1 ze{h2 Þ p’I MD ~
n 1ze{h1 ze{h2 n n, eE ð1ze{E{h1 ze{E{h2 Þ zð1ze{h1 ze{h2 Þ
ð27Þ
ð28Þ
ð29Þ
where the subscript MD stands for ‘‘Mono-Double’’. Double-receptor/Double-ligand (DD) properties at equilibrium. Under the same assumptions of the previous sections, any doublereceptor/double-ligand system, built by n receptors [i g (1, ¼, n)] and whose occupancy is ruled by si 5 (0, 1), and ti 5 (0, 1) can be described by the following allosteric Hamiltonian function H ðs,t,aÞ~H1 ðs,aÞzH2 ðt,aÞ ! n1 n2 n1 n2 X X X X si {E2 ti azh1 si zh2 ti : ~ 2E{E1 i~1
i~1
i~1
ð30Þ ð31Þ
i~1
We note that the system factorizes into two independent MonoMono Hamiltonians, hence we can entirely skip the calculations, referring to results already presented in the section devoted to Mono-receptors/Mono-ligand properties. Thus, focusing on a symmetric case for simplicity, i.e. E1 ~E2 ~E and n1 5 n2 5 n, we get ðpA ÞDD ~
n ð32Þ e{2E 1zeE{h1 zeE{h2 zeE{h1 eE{h2 , n n n {h {h {2E E{h E{h E{h E{h ð1ze 1 Þ ð1ze 2 Þ ze ð1ze 1 ze 2 ze 1 e 2 Þ
while, via the dual assumptions A’, we have 0 pA DD ~ n ð33Þ e2E 1ze{E{h1 ze{E{h2 ze{E{h1 e{E{h2 n n n, ð1ze{h1 Þ ð1ze{h2 Þ ze2E ð1ze{E{h1 ze{E{h2 ze{E{h1 e{E{h2 Þ where the subscript DD stands for ‘‘Double-Double’’. 5
www.nature.com/scientificreports Logical operations. Let us now explore the possibility of using these allosteric receptor-ligand systems as operators mimicking stochastic logic gates: the presence of ligands (variables in Logic) is denoted as Si for the i-th ligand, and the presence of receptors (operators in Logic) is denoted as RA,i and RI,i for the active and inactive state of the i-th receptor, respectively: note that si and Si are conceptually different because, in Logic, Si mirrors the presence of the i-th ligand, that is Si 5 ‘‘true’’ stands for a high concentration presence of the i-th ligand, thus within the statistical mechanical route the S’s are linked to the h’s rather than to the s’s. Operators are of two kinds: the unary operators YES and NOT, which evaluate a single argument, and the binary operators, e.g., AND and OR, which evaluate two arguments. Let us describe the examples of concrete interest in the paper: – Affirmation: ‘‘S’’, namely the evaluation of the presence of ligand S, which returns true if and only if the ligand S is present. Hereafter this operator will be denoted as stochastic YES (or, in case a distinction between several ligands is necessary, as YESS). – Negation: ‘‘:S’’, namely the evaluation of the absence of ligand S, which returns true if and only if the ligand S is not present. Hereafter this operator will be denoted as stochastic NOT (or NOTS). – Conjunction: ‘‘S1 ^ S2 ’’, namely the evaluation of the presence of both ligands, which returns ‘‘true’’ whenever both ligands occur to be present (i.e., in the case that S1 and S2 are assigned value ‘‘true’’) and ‘‘false’’ whenever at least one of the two ligands is not present (i. e., in the case that either S1 or S2 are assigned value false). The evaluation of such operator is hereafter denoted as S1 AND S2 (stochastic AND). – Non-exclusive disjunction: S1 _ S2 , namely the evaluation of the presence of at least one ligand, which returns true whenever at least one ligand is present and value false whenever they are both absent. The evaluation of such operator is hereafter denoted as S1 OR S2 (stochastic OR). As we will see, the receptor molecule plays as an operator, while ligands play as variables. In order to evaluate the formula, each variable can assume value either ‘‘true’’ of ‘‘false’’ according to the ligand concentration, where ‘‘true’’ means that the ligand is present at a concentration larger than a threshold value, while ‘‘false’’ means that the ligand concentration is smaller than such a value. Moreover, the value arising from the evaluation of the operators corresponds to the activation state of the receptor: active if the evaluation returns ‘‘true’’ and inactive is evaluation returns ‘‘false’’.
Mono-receptor/Mono-ligand system: YES and NOT functions. All the plots in this and in the following sections are based on some scaling assumptions that will be discussed further in the paper (see Section ‘‘Methods’’). These assumptions are essential to our purpose (that is, they enable us to tune the free variables introduced defining the Hamiltonians), and are deduced by physical and biochemical reasoning. We will refer to these assumptions as they are reported in Section Methods below. Under scaling assumptions (35), (39) and (40), plots of the activation probability (pA)MM(h) from eq. (12) show marked sigmoidal behavior (see Fig. 3, upper panel), signaling activation of the receptor in significative presence of the ligand, i.e. for small values of the variable h: the logarithmic relation between h and the concentration follows directly both from the original MWC model, as summarized in Table 1 and from the Thompson approach41. Thus, the function (pA)MM may be considered as mimicking the logical YES[L] function, assuming boolean values 0 for low ligand concentration and 1 for high ligand concentration, as one can see from Tab. 2. which can in turn be fixed by The threshold value is set at h properly choosing the system constituents (e.g. the number of binding sites hosted by a receptor). On the contrary, the function p’A MM of eq. (14) may be considered as mimicking the logical NOT[L] function (Fig. 3, lower panel), assuming boolean values 0 for high ligand concentration and 1 for low ligand concentration, as one can see from Tab. 2. A comparison of the theoretical versus real behavior of these gates is presented in Fig. 4, while the best fitting procedure is discussed in Section ‘‘Best fitting procedures’’. Mono-receptor/Double-ligand system: OR and NOR functions. The activation probability (pA)MD (eq. (26)) can be used to model a stochastic version of the logic gate OR. In fact, if we look at the presence of the two different ligands as a binary input, the behavior of (pA)MD (with the scaling assumptions of eqs. (35), (39)), as a function of h1 and h2 (see Fig. 5), recovers the OR’s one (see Tab. 2). Similarly to the YES case, the value 0 for h1 and for h2 denotes the saturation of the ligand. Therefore, consistently with the structure of OR, the presence of only one out of the two ligands is sufficient to make the molecule active; conversely, the value E denotes the absence, thus for h1 ~h2 ^E, (pA)MD is vanishing, namely, it returns as output ‘‘false’’. Note that the projection of the plot over
Figure 3 | Upper panel: Sigmoidal behavior of pA(h) with parameters E 5 2n, E~2E=n, where n 5 5 (blue), n 5 50 (red), n 5 500 (gold). Lower panel: Anti-sigmoidal behavior of p’A ðhÞ with parameters E 5 2n, E~2E=n, where n 5 5 (blue), n 5 50 (red), n 5 500 (gold). SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
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Table 2 | The truth table of all the logical operators introduced by now Input
YESA
NOTA
A OR B
A NOR B
A AND B
A NAND B
A
B
A
:A
A~B
A#B
A‘B
A"B
1 1 0 0
1 0 1 0
1 1 0 0
0 0 1 1
1 1 1 0
0 0 0 1
1 0 0 0
0 1 1 1
h1 ~E (or h2 ~E) gives a sigmoid, consistently with the fact that, if one of the two inputs is constantly false, the OR recovers the YES. Performing the same calculations, the dual counterpart p’A MD of eq. (28) models the logical NOR gate, that is the direct negation of the previous one, as shown in Fig. 5). A comparison of the theoretical versus real behavior of these gates is presented in Fig. 6 (for the OR only), whose data have been collected from an experiment sketched in Fig. 7 (lower panel); the best fitting procedure is discussed in the Section ‘‘Best fitting procedures’’. Double-receptor/Double-ligand system: AND and NAND functions. The function (pA)DD previously described (eq. (32)) models a stochastic version of the logic AND gate (see Tab. 2). As in the case of OR, we look at the two ligands as a binary input, and we assume the scaling assumptions coded in eqs. (35), (42), (44). The resulting behavior of (pA)DD fits the one expected for the AND function, with fitness to the expected plot that sensibly improves in the extremal regions of the plot, i.e. for h1,2 *0,E (see Fig. 8). Again, its projection returns a sigmoid because if one of the two inputs is constantly true, the AND recovers the YES. The dual version p’A DD (eq. (33)) models the logic gate NAND, i.e. the direct negation of the previous one. As this negation is precisely dual, so is the shape of the plot (see Fig. 8). A comparison of the theoretical versus real behavior of these gates is presented in Fig. 9 (for the AND only), while, mirroring the exposition of the OR gate, Fig. 7 (upper panel) summarizes the experiment working as data source; the best fitting procedure is discussed in the Section ‘‘Best fitting procedures’’.
Conclusions Chemical computing uses molecular systems to perform logical operations, mimicking processes typical of electronic devices. Advantages and disadvantages appear when comparing these two approaches to computation: chemical computing requires (for a single operation) a smaller size (Angstroms versus microns) and a lower energy consumption (,10219 Joule versus , 1029 Joule), yet it is slower than electronic computing (from kilohertz to megaherz versus gigahertz). Further, biochemical information processing performs at
relatively large levels of noise (and this happens at different scales, ranging from enzyme-based logic gates10 to nucleic acid logic circuits21: noise propagates in the system as thermal disorder or in form of cross-talk among system’s constituents4,49. Hence, classical (i.e. deterministic) logic works only as an ideal reference framework and the field of research would strongly benefit from a robust stochastic reformulation of logic gates whose properties can be safely tested and used as guides in in-vitro experiments. In this work we showed that statistical mechanics can play a major role to accomplish this task: through statistical-mechanical models, we analyzed paradigmatic allosteric reaction kinetics and we proved their ability in performing noisy operations (hence working as suitable versions of stochastic logic gates). Moreover, we tested their performances (as resulting from the theory) on real data taken from enzyme-based information processing systems finding overall very good agreement between theoretical predictions and experimental behaviors. From this perspective, we showed that allosteric kinetics of mono receptor systems naturally encodes a noisy version of the input-output relations typical of the YES and NOT gates, while allosteric kinetics of double receptors systems plays as a stochastic version of the OR (NOR) and AND (NAND) gates. Finally, we checked that in the noiseless limit of the theory (for n R ‘, or, alternatively, for b R ‘) these gates recover the pure logic-like behavior (i.e. deterministic information processing). The framework developed here allows establishing controllable, explicit relations among the system parameters, with remarkable practical outcomes. For instance, the noise that generally affects biological gates can be quantified permitting the management of large devices where different gates work in cascade and where noise-driven errors (possibly amplified by the various gates) actually constitute severe limitations. Also, in order to build large biochemical circuits, understanding how simple building blocks behave and how their interactions scale up with system’ size is mandatory as well as quantifying the computational power of the system itself: classical reaction equations (i.e. systems of differential equations) can not accomplish these tasks, while statistical mechanics can. In fact, combining Hamiltonians coding for various gates in a cascade fashion is straightforward in the present formalization, further, an arsenal of
Figure 4 | Left: (pA)DD(h1, h2) plots. Activation of the receptor isverified by small values of h1 and h2, corresponding to a significative presence of both the two ligands, thus simulating a stochastic AND function. Right: p’A MD ðh1 ,h2 Þ plots. Activation of the receptor is verified by high (i.e. small in absolute value) values of h1 or h2, corresponding to a significative presence of any of the two ligands, thus simulating a stochastic NAND function. SCIENTIFIC REPORTS | 5 : 9415 | DOI: 10.1038/srep09415
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Figure 5 | The stochastic OR gate has been realized in two coupled steps involving enzymatic processes as sketched in Fig. 7: first enzyme is esterase, that reacts with ethyl butyrate ([A]) or methyl butyrate ([B]), or both, catalyzing production of ethanol and methanol, respectively. Butyric acid is a byproduct of the process. To set the gate, the physical zeros of the signals have been fixed experimentally to convenient input values (ethyl butyrate 10 mM and methyl butyrate 10 mM). Bullets represent experimental data8, whereas the surface represents the best fitting according to eq. (26).
tools stemmed from the neural network branch (e.g., methods to address system’s computational capacity, memory storage, etc.27) becomes immediately handily via this route.
Methods In this section we discuss three major aspects of our work: the scaling assumptions, the role of allosteric cooperativity within the model and the best fitting procedures. Scaling assumptions. As the assumption sets A and A’ only affect the sign of the parameters E, E and of the variable h, we cannot expect every choice of these quantities to yield a realistic behavior from a biophysical viewpoint. Particularly an effective range of the variable h as well as some reasonable scaling properties for E and E are to be determined, most likely depending on n. The first issue can be solved independently of the case considered (MM, MD, DD). As evidenced in Tab. 1, for assumptions A it is e2h 5 [S]/KI and, being h positive, activation enhancement [S]/KI is dimensionless and ranging in [0, 1], thus, it may be considered as a percent molar concentration of the ligand S. Also, we expect that there exists a numerical (percent) value for the ligand concentration, below which the receptor activity is unaffected (see e.g. Ref. 46). We refer to this threshold value as t and, according to Tab. 1, this also determines the significance range of h as 0vhv{logt,
ð35Þ
Dually, for assumptions A’ it is eh 5 KI/[S] and the same conclusion follows that KI/[S] may be considered as a percent molar concentration of the ligand S. As for t9 we have t9 < KI/KA (following from assumptions (c9), (e9)), yielding
ð36Þ
Now we focus on the scaling of E and E: in the following we address this matter separately for the case of one or two receptors, which have different nature. Mono-receptor case: YES/NOT and OR/NOR gates. We refer only to assumptions A, since dual gates clearly scale in the same way. Let us start considering the Mono-Mono case:given Eq. 12, we can define h as the value of the dissociation energy ~1=2, which implies such that ðpA ÞMM h n n e{E 1zeE{h ~ 1ze{h : ð37Þ On the other hand, the active (a 5 1) and saturated (s 5 1) state is an extremal state of the system corresponding to minimum entropy. As a result ~E{nEz H a~1,s~1,h hn~0: ð38Þ From the previous two equations we have E, h~ n
ð34Þ
which reliably limits the range of the dissociation energy to finite values. As t determines the receptor sensitivity with respect to its activity, it is reasonably expected that t < KA/KI; in fact such inverse proportional dependence of t with respect to KI is consistent with increasing monotonicity of h with respect to KI (consistently with assumptions (c), (e)). Moreover, from Tab. 1, t
